{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:32:16Z","timestamp":1787340736465,"version":"build-2736575974"},"reference-count":50,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2024,12,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We consider the linear conic optimization problem with the cone of nonnegative polynomials. Its dual optimization problem is the generalized moment problem. Moment-sum of squares (SOS) relaxations are powerful for solving them. This paper studies finite convergence of the Moment-SOS hierarchy when the constraining set is defined by equations whose ideal may not be real radical. Under the archimedeanness, we show that the Moment-SOS hierarchy has finite convergence if some classical optimality conditions hold at every minimizer of the optimal nonnegative polynomial for the linear conic optimization problem. When the archimedeanness fails (this is the case for unbounded sets), we propose a homogenized Moment-SOS hierarchy and prove its finite convergence under similar assumptions. Furthermore, we also prove the finite convergence of the Moment-SOS hierarchy with denominators. In particular, this paper resolves a conjecture posed in the earlier work.<\/jats:p>","DOI":"10.1137\/23m1605430","type":"journal-article","created":{"date-parts":[[2024,10,3]],"date-time":"2024-10-03T04:00:27Z","timestamp":1727928027000},"page":"3399-3428","source":"Crossref","is-referenced-by-count":8,"title":["Finite Convergence of Moment-SOS Relaxations with Nonreal Radical Ideals"],"prefix":"10.1137","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9734-5387","authenticated-orcid":true,"given":"Lei","family":"Huang","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of California at San Diego, La Jolla, CA 92093 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiawang","family":"Nie","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of California at San Diego, La Jolla, CA 92093 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ya-Xiang","family":"Yuan","sequence":"additional","affiliation":[{"name":"Institute of Computational Mathematics and Scientific\/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190 China."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,10,3]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-022-01877-6"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/054"},{"key":"ref3","volume-title":"Nonlinear Programming","author":"Bertsekas D.","year":"1995","edition":"2"},{"key":"ref4","doi-asserted-by":"crossref","unstructured":"D. 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